Transcript Problem 11 - Texas A&M University

Chapter 12: Gravitation
Newton’s Law of Gravitation
Gm1m2
Fg 
r2
G=gravitational constant = 6.673(10) 10-11 Nm2 / kg 2
Note: The weight  of a body of mass m on the earth's surface with
radius R E is
GmE  m
GmE
  mg 
2
E
R
or g 
2
E
R
Gravitational attraction
Note: Two particles of different mass exert equally
strong gravitational force on each other
Gravitational Forces (I)
MM ME
MM
FG = G
r2
‘‘Attractive Force”
ME
Cavendish balance
Cavendish(1798) announced that he has weighted the earth
Average Density of the Earth
g = 9.80 m/s2
RE = 6.37 x 106 m
ME = 5.96 x 1024 kg
 rE = 5.50 x 103 kg/m3
= 5.50 g/cm3 ~ 2 x rRock
 ???
The gravitational force is conservative
Note:
Work done is
Path independent!
Gravitational potential energy
Note: The gravitational potential is zero at infinity
Projectile
K1  U1  K 2  U 2
GmE m
GmE m
1 2
mv1  (
)  0  (
)
2
RE
2 RE
v1  GmE / RE
Projectile
Circular orbit
G  mE  m mv

2
r
r
v  GmE / r
2
Kepler’s Laws
1. Each planet moves in an elliptical orbit,
with the sun at one focus of the ellipse
2. A line from the sun to a given planet
sweeps out equal areas in equal times.
3. The periods of the planets are
proportional to the 3/2 powers of the major
axis lengths of their orbits.
Geometry of an ellipse
Elliptical orbit
dL
  r  F  0
dt